MATH-UA.212 Written HW5 Write full, clear solutions to the problems below. It is important that the logic of how you solved these problems is clear. Although the final answer is important, being able to convey you understand the underlying concepts is more important. Be sure to write your name and recitation section on your homework, and be sure to staple all loose pages. \u0014 \u0015 3 0 1 1. (2 points) Find A if A = . 2 1 2. Data interpolation has many applications. Suppose we have been given the task of finding the degree 3 polynomial P (x) = ax3 + bx2 + cx + d which satisfies P 0 (1) = 3 and passes through the points (1, 1), (0, 1), and (1, 3). (a) (3 points) Show that the coefficients of P satisfy the system of equations 3a 2b + c = 3 a + b c + d = 1 d=1 a + b + c + d = 3. a b Write a matrix equation of the form A c = r so we could solve for a, b, c, and d. d (b) (5 points) Find A1 , and use it to find a, b, c, and d. Write the formula for P (x). (c) (2 points) Your boss suddenly changes the problem at the last minute. You have now been asked to find the degree 3 polynomial f which satisfies f 0 (1) = 13 and passes through the points (1, 3), (0, 2), and (1, 3). Use your work from parts (a) through (c) to find the formula for f . 3. Find the following determinants: 1 1 0 (a) (2 points) 1 3 2 1 0 0 a b c (b) (2 points) 0 d e 0 0 f 4. (3 points) Find all values of x so that the following matrix is invertible: 2x 0 3 0 1 2x 0 3 . A= 0 0 2x 0 0 0 1 2x 5. (5 points) Find 1 0 0 3 0 2 1 0 6 0 3 2 4 9 1 4 1 4 2 0 0 . 12 4 1 1 6. (4 points) If A and B are invertible n n matrices, show that the inverse of AB is B 1 A1 . In other words, show (AB)1 = B 1 A1